Hostname: page-component-7c8c6479df-r7xzm Total loading time: 0 Render date: 2024-03-28T02:49:28.884Z Has data issue: false hasContentIssue false

On Springer's correspondence for simple groups of type En (n = 6, 7, 8)

Published online by Cambridge University Press:  24 October 2008

Dean Alvis
Affiliation:
Massachusetts Institute of Technology
George Lusztig
Affiliation:
Massachusetts Institute of Technology

Extract

Let G be a connected reductive algebraic group over complex numbers. To each unipotent element u ε G (up to conjugacy) and to the unit representation of the group of components of the centralizer of u, Springer (11), (12) associates an irreducible representation of the Weyl group W of G. The tensor product of that representation with the sign representation will be denoted ρu. (This agrees with the notation of (5).) This representation may be realized as a subspace of the cohomology in dimension 2β(u) of the variety of Borel subgroups containing u, where β(u) = dim . For example, when u = 1, ρu is the sign representation of W. The map u → ρu defines an injective map from the set of unipotent conjugacy classes in G to the set of irreducible representations of W (up to isomorphism). Our purpose is to describe this map in the case where G is simple of type Eu (n = 6, 7, 8). (When G is classical or of type F4, this map is described by Shoji (9), (10); the case where G is of type G2 is contained in (11).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Alvis, D. Induce/restrict matrices for exceptional Weyl groups. (Manuscript.)Google Scholar
(2)Barbasch, D. & Vogan, D. Primitive ideals and orbital integrals in complex exceptional groups. (Manuscript.)Google Scholar
(3)Beynon, W. M. & Lusztig, G.Some numerical results on the characters of exceptional Weyl groups. Math. Proc. Cambridge Phil. Soc. 84 (1978), 417426.CrossRefGoogle Scholar
(4)Borho, W. & Macpherson, R. Intersection homology of nilpotent varieties. (Manuscript.)Google Scholar
(5)Lusztig, G.A class of irreducible representations of a Weyl group. Proc. Nederl. Akad., series A, 82 (3), 1979, 323335.Google Scholar
(6)Lusztig, G.Some problems in the representation theory of finite Chevalley groups. Proc. Symp. Pure Math. 37 (1980).Google Scholar
(7)Lusztig, G. Green polynomials and singularities of unipotent classes. Advances of Math. (in the Press).Google Scholar
(8)Lusztig, G. & Spaltenstein, N.Induced unipotent classes. J. London Math. Soc. 19, 1979, 4152.CrossRefGoogle Scholar
(9)Shoji, T.On the Springer representations of the Weyl groups of classical algebraic groups. Comm.in Alg. 7 (1979), 1713–1745, 20272033.CrossRefGoogle Scholar
(10)Shoji, T.On the Springer representations of Chevalley groups of type F4. Comm. in Alg. 8 (1980), 409440.CrossRefGoogle Scholar
(11)Springer, T. A.Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math. 36 (1976), 173207.CrossRefGoogle Scholar
(12)Springer, T. A.A construction of representations of Weyl groups. Invent. Math. 44 (1978), 279293.CrossRefGoogle Scholar